How to Reset a Previously Owned iOS Device

I got a fourth-gen iPod touch off of eBay. I have it named and have an Apple ID. 
But, when I go to the App Store or Game Center, etc., the previous owner’s Apple 
ID shows up and it asks for a password. How do I clean this up and get my Apple ID to show up?

The best way to “clean up” a device that was previously owned is to reset iOS and all of the settings to start anew. This will wipe the device of all of the previous owner’s settings and content, allowing you to enter your information as if the device were new.